Intereting Posts

Sign of determinant when using $det A^\top A$
Subordinate matrix norm
Extreme Value Theorem Proof (Spivak)
Finding all complex zeros of a high-degree polynomial
The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.
Inversion of Trigonometric Equations
Sample variance converge almost surely
Proving $\ell^p$ is complete
Find a non-negative function on such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable
What is an intuitive explanation for $\operatorname{div} \operatorname{curl} F = 0$?
If $T(n+1)=T(n)+\lfloor \sqrt{n+1}, \rfloor$ $\forall n\geq 1$, what is $T(m^2)$?
Simplifying an integral by changing the order of integration
Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?
Integral $\int_0^{\large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx$
What is the value of $\int_0^1 \frac{\arctan x}{1+x^{2}} dx$?

I’m writing an article on Lychrel numbers and some people pointed out that this is completely useless.

My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.

I came with some ideas like the Turing machine but I think I’m not grasping the right examples.

- When do we use Tensor?
- Why does GPS require a minimum of 24 satellites?
- How does linear algebra help with computer science
- Applications of Gröbner bases
- Is there any abstract theory of electrical networks?
- Applications of algebraic topology

Can someone point me some theories that seemed like the Lychrel numbers and then become ‘useful’?

- Which calculus text should I use for self-study?
- Good Physical Demonstrations of Abstract Mathematics
- Where to begin in approaching Stochastic Calculus?
- Quotient geometries known in popular culture, such as “flat torus = Asteroids video game”
- Applications of the number of spanning trees in graphs
- Sum of digits and product of digits is equal (3 digit number)
- “Long-division puzzles” can help middle-grade-level students become actual problem solvers, but what should solution look like?
- What are some surprising appearances of $e$?
- Why do we need to prove $e^{u+v} = e^ue^v$?
- Easy example why complex numbers are cool

The Quaternions were considered useless for a long time.

Anyhow, the set of all unit quaternions is a double cover of $SO_3(\mathbb R)$. This allows us to represent any rotation matrix by a quaternion, which is used now in computer games (instead of using 9 parameters to parametrize a rotation matrix, we can use only 3 for the quaternions).

You can read more here.

Quote from G. H. Hardy^{1}

The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics.

Just 30 years after his death, the RSA algorithm was introduced which is deeply rooted in number theory and is now important part of sending encrypted information electronically, e.g., over the Internet.

The solutions of the closest-packing problem, of how to most densely pack non-overlapping congruent $n$-spheres in $\mathbb R^n,$ have, for some $n>3,$ been found to have applications to error-detecting and error-correcting codes in digital transmissions.

In the book Knots by Kaufman there is an example of an application of knot theory (the study of homeomorphic embeddings of $S^1$ into $\mathbb R^3$) to statistical mechanics.

- Number of distinct numbers picked after $k$ rounds of picking numbers with repetition from $$
- Lebesgue integral basics
- When is the topological closure of an equivalence relation automatically an equivalence relation?
- Show a convergent series $\sum a_n$, but $\sum a_n^p$ is not convergent
- A series with only rational terms for $\ln \ln 2$
- Why are measures real-valued?
- Convergence of $x_{n+1} = \frac12\left(x_n + \frac2{x_n}\right).$
- Cross product of cohomology classes: intuition
- Fractals using just modulo operation
- Linear algebra – find all possible positions of the third corner?
- Proving :$\arctan(1)+\arctan(2)+ \arctan(3)=\pi$
- Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
- cusp and node are not isomorphic
- Why does the tangent of numbers very close to $\frac{\pi}{2}$ resemble the number of degrees in a radian?
- Trace of a nilpotent matrix is zero.